Implicit adaptive mesh refinement for 2D reduced resistive magnetohydrodynamics

Philip, Bobby; Chacòn, Luis; Pernice, Michael

doi:10.1016/j.jcp.2008.06.029

Cited by 21 publications

(39 citation statements)

References 60 publications

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“…Block-structured adaptive mesh refinement methods discretize PDEs with a hierarchy of regular grids to adaptively refine around regions of interest [3,4,38]. For such problems, fast adaptive composite linear solvers are often used, which employ fast uniform-grid algorithms on each level iteratively to achieve the solution on the hierarchical composite mesh, and have proven beneficial in radiation diffusion and reduced MHD applications [30,32,33]. This preconditioning approach could easily be utilized in such methods.…”

Section: Discussionmentioning

confidence: 99%

Operator-Based Preconditioning of Stiff Hyperbolic Systems

Reynolds¹,

Samtaney²,

Woodward³

2010

SIAM J. Sci. Comput.

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Abstract.We introduce an operator-based scheme for preconditioning stiff components encountered in implicit methods for hyperbolic systems of PDEs posed on regular grids. The method is based on a directional splitting of the implicit operator, followed by a characteristic decomposition of the resulting directional parts. This approach allows for the solution of any number of characteristic components, from the entire system to only the fastest, stiffness-inducing waves. We apply the preconditioning method to stiff hyperbolic systems arising in magnetohydrodynamics and gas dynamics. We then present numerical results showing that this preconditioning scheme works well on problems where the underlying stiffness results from the interaction of fast transient waves with slowly-evolving dynamics, scales well to large problem sizes and numbers of processors, and allows for additional customization based on the specific problems under study.

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Section: Discussionmentioning

confidence: 99%

Operator-Based Preconditioning of Stiff Hyperbolic Systems

Reynolds¹,

Samtaney²,

Woodward³

2010

SIAM J. Sci. Comput.

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“…Another interesting challenge in developing implicit methods for MHD is the combination of JFNK or FAS methods with adaptive mesh refinement (AMR). Some progress towards JFNK with AMR has been reported by Philip et al (2008) on reduced incompressible MHD in 2D. Combining implicit methods with AMR will help mitigate not only the temporal stiffness issues but also help effectively resolve the range of spatial scales in MHD.…”

Section: Future Challengesmentioning

confidence: 99%

Implicit Numerical Methods for Magnetohydrodynamics

Samtaney¹

2012

Topics in Magnetohydrodynamics

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“…In this section, we investigate several MHD test problems to show that the nested iteration Newton-FOSLS-AMG method is capable of solving complex nonlinear systems in about 30-80 work units or fine-grid-relaxation equivalents. The full algorithm, as in figure 3.4, was applied to two tokamak test problems [13,14,23,27]. From the papers by Chacón, Knoll, and Finn [13] and Philip [23], a reduced set of MHD equations is obtained that simulate a "large aspect-ratio" tokamak with non-circular cross-sections.…”

Section: The Mhd Equations and Fosls Formulationmentioning

confidence: 99%

“…The full algorithm, as in figure 3.4, was applied to two tokamak test problems [13,14,23,27]. From the papers by Chacón, Knoll, and Finn [13] and Philip [23], a reduced set of MHD equations is obtained that simulate a "large aspect-ratio" tokamak with non-circular cross-sections. Here, the magnetic B-field along the z-direction, or the toroidal direction, is very large and mostly constant.…”

Section: The Mhd Equations and Fosls Formulationmentioning

confidence: 99%

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Nested Iteration and First-Order System Least Squares for Incompressible, Resistive Magnetohydrodynamics

Adler¹,

Manteuffel²,

McCormick³

et al. 2010

SIAM J. Sci. Comput.

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Abstract. This paper develops a nested iteration algorithm to solve time-dependent nonlinear systems of partial differential equations. For each time step, Newton's method is used to form approximate solutions from a sequence of nested spaces, where the resolution of the approximations increases as the algorithm progresses. Nested iteration results in most of the iterations being performed on coarser grids, where minimal work is needed to reduce error to the level of discretization error. The approximate solution on a given coarse grid is interpolated to a refined grid and used as an initial guess for the problem posed there. The approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem, due to the rapid convergence of Newton's method near a solution. The paper develops an algorithm that attempts to optimize accuracy-percomputational-cost on each grid, so that essentially no unnecessary work is done on any grid. The nested iteration algorithm is then applied to a reduced 2D model of the incompressible, resistive magnetohydrodynamic (MHD) equations. Using this algorithm on the MHD equations in the context of a first-order system least-squares (FOSLS) finite element discretization and algebraic multigrid (AMG) to solve the linearized systems, instabilities in a model tokamak fusion reactor are simulated. Numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30-80 fine-grid relaxation sweeps per time step.Key words. magnetohydrodynamics, algebraic multigrid, FOSLS, nested iteration AMS subject classifications. 65F10, 65M55, 76W05Acknowledgments. This work was sponsored by the Department of Energy under grant numbers DE-FG02-03ER25574 and DE-FC02-06ER25784, Lawrence Livermore National Laboratory under contract numbers B568677, and the National Science Foundation under grant numbers DMS-0621199, DMS-0749317, and DMS-0811275.1. Introduction. This paper applies a nested iteration algorithm to a set of incompressible resistive Magnetohydrodynamic (MHD) equations. A first-order systems least-squares (FOSLS) [11,12] finite element discretization is used. The main focus of this paper is to show that nested iteration is a crucial component to solving complicated systems of nonlinear partial differential equations. As is shown for the MHD equations, nested iteration allows for a complicated system of nonlinear equations to be solved to within discretization accuracy in only a handful of work units (WUs). A WU is defined as the amount of computation required to perform one fine-grid relaxation sweep. The main idea is that most of the work is done on coarse grids where computation is much cheaper. The approximation to the solution on these coarse grids is interpolated up to successively finer grids, where linearizations and algebraic solves are applied. The process is then continued to finer grids until a desired error tolerance or resolution is reached. The result is that coarse grids give fairly...

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Implicit adaptive mesh refinement for 2D reduced resistive magnetohydrodynamics

Cited by 21 publications

References 60 publications

Operator-Based Preconditioning of Stiff Hyperbolic Systems

Operator-Based Preconditioning of Stiff Hyperbolic Systems

Implicit Numerical Methods for Magnetohydrodynamics

Nested Iteration and First-Order System Least Squares for Incompressible, Resistive Magnetohydrodynamics

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